The news that justice for Stephen Lawrence has been achieved has left me ecstatic for this family – and speechless. The verdict sends out shockwaves: it inspires all those who had given up hope, and could possibly signal a new beginning in our society.

The most important thing is that it will bring closure to the family; but from a wider perspective the police and criminal justice system should see this as creating a new baseline for establishing standards of investigation and prosecution for all hate crimes.

That depends on how we create a morality in society where we preach the virtues of respect and dignity – which in turn lies with our politicians, our media, our educators and those who influence opinion. They have got to give leadership in challenging ignorance and bigotry and suppressing their inclinations towards demonising people.

The Stephen Lawrence campaign was the most momentous race campaign I have known. It endured because for black communities it symbolised the failures of the police, which they felt in their lives every day.

On 22 April 1993, when Stephen Lawrence was killed, "living in fear" was an apt description for the climate of race relations. Just that week I had become chairman of the Commission for Racial Equality. I got a phone call telling me that a black boy had been the victim of a racially motivated murder, and that all hell could break loose.

Many good people came forward immediately to support the family. Ros Howells, a local activist and community stalwart, provided a solid rock of continuous support to the family, and the local MP, Peter Bottomley, stepped forward, as did many other well-wishers. But above all, Stephen's parents, Doreen and Neville Lawrence – helped by their family and close friends – became a formidable force in their dedicated pursuit of justice for their son.

Eltham was one of the "no-go" areas in London for black people in the dark days of the early 90s, and it was no surprise to me, as a south London resident, that such a brutal murder should occur there. Racist and fascist groups were active, and their racial terrorism took the form of abuse, threats, and petrol bombs and excrement through letterboxes or windows. Many families – harassed on the streets, and their children abused at school – were afraid to go out alone.

The murder of Stephen Lawrence was a defining moment for the way in which black victims of racially motivated crime were to be seen and responded to. There was little or no confidence that the police or courts took such crimes seriously, and a black life seemed to have little value. The Lawrence family showed the way forward in conducting themselves with dignity, demonstrating the integrity and aspirations of their well-educated son, and highlighting the inadequacies of the authorities in their unprofessional and unsympathetic responses to his murder.

So impressive was their unwavering attempt to get justice for Stephen that they gained the support of the Daily Mail – not known for backing black causes – which expressed its own outrage at the failures of the police, most famously by naming the five main suspects as "Murderers" in a dramatic front-page headline.

In 1997 Jack Straw, home secretary in the newly elected Labour government, set up the Macpherson inquiry. The hearings uncovered a litany of complacency and incompetence among senior police officers who had initially investigated the Lawrence murder. When it reported in February 1999, among its findings was a clear conclusion about the existence of institutional racism in the police service and other public sector organisations. It was a vindication of those campaigners who had been vilified for demanding stronger equality and anti-discrimination laws.

In reaction to the report, some politicians, senior officers and public service chiefs admitted wrongdoing and promised to change, while others rubbished the report by insisting, ludicrously, that it branded all white people in institutions as individually racist.

But since Macpherson there has been significant progress in making Britain a fairer and more just society than in the 1990s. The "Lawrence moment" inspired many individuals and organisations to collectively tackle racially motivated abuse, crime and violence; to raise awareness of racial prejudice and bigotry; and to campaign against institutional discrimination, which was rife in all the apparatus of state, including the judiciary, parliament and police, as well as many private institutions.

The report led to the introduction of the Race Relations (Amendment) Act of 2000. Sir Charles Guthrie, then head of the armed services, declaring "war" on racism and sexism, expressed his determination to see the army, navy and RAF become inclusive and free from discrimination. He was backed by the defence secretary, John Reid. Such declarations sent out powerful messages.

Similar leadership was given on policing by Sir John Stevens, the Metropolitan police commissioner from 2000-2005, who made it clear he was going to tackle inefficiency, incompetence and corruption, and seeking improved police/community relationships built on mutual trust and confidence. His bold and open-minded approach represented a fundamental rejection of the previous self-denial and defensive attitudes prevalent in many public bodies, but especially in the police service. It was no surprise to see more people from minority ethnic communities wanting to join the police and to help them prevent and solve crime.

There was even a commitment by the lord chancellor to seek the appointment of minorities and women to the high ranks of the judiciary.

Such leadership gave credence to the notion that Britain had woken up to the evils of racism and was serious about tackling it. Black and minority ethnic people were becoming more confident and assertive, believing they could help the country become free from racial inequality, attacks, harassment, abuse and discrimination. Visible symbols of political successes in high-powered positions included Paul Boateng and Valerie Amos in Tony Blair's cabinet.

There have been setbacks since: most notably in the targeting of Muslims over the "war on terror", and the onslaught against multiculturalism. But there remain some enlightened individuals in leadership positions who are prepared to seek a fairer society for all.

Stephen Lawrence happened to be in the wrong place at the wrong time in April 1993 simply because of the colour of his skin. His family still grieve his loss. But Doreen and Neville Lawrence, in their unrelenting and now successful campaign for justice, can be assured that they have inspired many others to challenge injustice, discrimination, indifference, hatred and institutional exclusion.

They are responsible for the conviction of the killers in some of Britain's highest-profile murders, including those of Stephen Lawrence, Damilola Taylor and Rachel Nickell. What's the secret of their success?